Differential Equations and Dynamical Systems

, Volume 27, Issue 4, pp 585–600 | Cite as

Entanglement Dynamics of a Three-level Atom in a Momentum Eigenstate Interacting with Non-linear Effect

  • N. H. Abd El-Wahab
  • Ahmed SalahEmail author
  • A. S. Abdel Rady
  • Abdel-Nasser A. Osman
Original Research


We consider a general Hamiltonian for a system which consists of a three level lambda configuration atom interacting with a one-mode cavity field. Besides the intensity-dependent coupling the model includes multi-photon process as well as a non-linear Kerr-Like medium effect. Furthermore, the atom and the field are assumed to be coupled with modulated coupling parameter which depends explicitly on time. The atom is initially prepared in a superposition state and field in a coherent state. Under a rotating wave approximation where fast oscillations are ignored, an exact solution for the wave function in Schrödinger equation is obtained. The momentum increment, the momentum diffusion and the field entropy are calculated. The results shown that in existence of the time dependent coupling parameter leads to a time delaying in the interaction which is twice the delay time for the independent case. The general conclusions reached are illustrated by numerical results.


Three-level atom The effective time-dependent coupling parameter Entropy Entanglement 


  1. 1.
    Nielsen, M., Chuang, I.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
  2. 2.
    Sainz, I., Klimov, A.B., Roa, L.: Entanglement dynamics modified by an effective atomic environment. Phys. Rev. A 73, 032303 (2006)CrossRefGoogle Scholar
  3. 3.
    Yuan, X.-Z., Goan, H.-S., Zhu, K.-D.: Non-Markovian reduced dynamics and entanglement evolution of two coupled spins in a quantum spin environment. Phys. Rev. B 75, 045331 (2007)CrossRefGoogle Scholar
  4. 4.
    Bradler, K., Jauregui, R.: Entanglement enhancement and postselection for two atoms interacting with thermal light. J. Phys. B 40, 743 (2007)CrossRefGoogle Scholar
  5. 5.
    Phoenix, S.J.D., Knight, P.L.: Establishment of an entangled atom-field state in the Jaynes-Cummings model. Phys. Rev. A 44, 6023 (1991)CrossRefGoogle Scholar
  6. 6.
    Phoenix, S.J.D., Knight, P.L.: Comment on “Collapse and revival of the state vector in the Jaynes-Cummings model: an example of state preparation by a quantum apparatus”. Phys. Rev. Lett. 66, 2833 (1991)CrossRefGoogle Scholar
  7. 7.
    Phoenix, S.J.D., Knight, P.L.: Fluctuations and entropy in models of quantum optical resonance. Ann. Phys. (N. Y.) 186, 381 (1988)CrossRefGoogle Scholar
  8. 8.
    Bennett, C.H., Di-Vincenzo, D.P., Smolin, J.A., Wootters, W.K.: Mixed-state entanglement and quantum error correction. Phys. Rev. A 54, 3824 (1996)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Vedral, V., Plenio, M.B., Rippin, M.A., Knight, P.L.: Quantifying entanglement. Phys. Rev. Lett. 78, 2275 (1997)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Abdel-Aty, M.: Influence of a Kerr-like medium on the evolution of field entropy and entanglement in a three-level atom. J. Phys. B At. Mol. Opt. Phys. 33, 2665 (2000)CrossRefGoogle Scholar
  11. 11.
    Furuichi, S., Ohya, M.: Entanglement degree in the time development of the Jaynes-Cummings model. Lett. Math. Phys. 49, 279 (1999)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Von Neumann, J.: Mathematische Grundlagen der Quantenmechanik. Springer, Berlin (1932)zbMATHGoogle Scholar
  13. 13.
    Cummings, F.W.: Stimulated emission of radiation in a single mode. Phys. Rev. 140, A1051 (1965)CrossRefGoogle Scholar
  14. 14.
    Sargent III, M., Scully, M.O., Lamb Jr., W.E.: Laser Physics. Addison-Wesley, Reading (1974)Google Scholar
  15. 15.
    Yoo, H.I., Eberly, J.H.: Dynamical theory of an atom with two or three levels interacting with quantized cavity fields. Phys. Rep. 118, 241 (1985)CrossRefGoogle Scholar
  16. 16.
    Li, X.S., Peng, Y.N.: Quantum properties of a three-level atom interacting with two radiation fields. Phys. Rev. A 32, 1501 (1985)CrossRefGoogle Scholar
  17. 17.
    Li, X.S., Gong, C.D.: Coherent properties of the stimulated emission from a three-level atom. Phys. Rev. A 33, 2801 (1986)CrossRefGoogle Scholar
  18. 18.
    Lai, W.K., Buzek, V., Knight, P.L.: Dynamics of a three-level atom in a two-mode squeezed vacuum. Phys. Rev. A 44, 6043 (1991)CrossRefGoogle Scholar
  19. 19.
    Peng, J.S., Li, G.X.: A study on dissipation mechanism in To-Photon laser. Acta. Phys. Sin. 41, 1590 (1992)Google Scholar
  20. 20.
    Kocharovskaya, O., Khanin, Y.I.: Population trapping and coherent bleaching of a three-level medium by a periodic train of ultrashort pulses. Sov. Phys. JETP 63, 945 (1986)Google Scholar
  21. 21.
    Boller, K.-J., Imamoglu, A., Harris, S.E.: Observation of electromagnetically induced transparency. Phys. Rev. Lett. 66, 2593 (1991)CrossRefGoogle Scholar
  22. 22.
    Fleischhauer, M., Imamoglu, A., Marangos, J.P.: Electromagnetically induced transparency: optics in coherent media. Rev. Mod. Phys. 77, 633 (2005)CrossRefGoogle Scholar
  23. 23.
    Scully, M.O., Zhu, S.-Y., Gavrielides, A.: Degenerate quantum-beat laser: lasing without inversion and inversion without lasing. Phys. Rev. Lett. 62, 2813 (1989)CrossRefGoogle Scholar
  24. 24.
    Scully, M.O., Zubairy, M.S.: Quantum Optics. Cambridge University Press, Cambridge (1997)CrossRefGoogle Scholar
  25. 25.
    Sandhya, S.N., Ravishankar, V.: Tomography, control, and characterization of entanglement in a three-level atomic system. Phys. Rev. A 82, 062301 (2010)CrossRefGoogle Scholar
  26. 26.
    Li, X-s: Zhu, S.-Y.: A generalized statistical N-level J-C model. Phys. A 131, 575 (1985)CrossRefGoogle Scholar
  27. 27.
    Lousiall, W.H., Yariv, A., Siegman, A.E.: Quantum fluctuations and noise in parametric processes. I. Phys. Rev. 124, 1646 (1961)CrossRefGoogle Scholar
  28. 28.
    Wodkiewicz, K.: Stochastic incoherences of optical Bloch equations. Phys. Rev. A 19, 1686 (1979)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Joshi, A.: Effects of phase fluctuations in the atom-field coupling coefficient of the Jaynes-Cummings model. J. Mod. Opt. 42, 2561 (1995)CrossRefGoogle Scholar
  30. 30.
    Abdel-Aty, M., Abdalla, M.S., Obada, A.-S.F.: Uncertainty relation and information entropy of a time-dependent bimodal two-level system. J. Phys. B At. Mol. Opt. Phys. 35, 4773 (2002)CrossRefGoogle Scholar
  31. 31.
    Abdalla, M.S., Obada, A.S.F., Abdel-Aty, M.: Entropy and entanglement of time dependent two-mode Jaynes–Cummings model. Phys. A 326, 203 (2003)MathSciNetCrossRefGoogle Scholar

Copyright information

© Foundation for Scientific Research and Technological Innovation 2016

Authors and Affiliations

  • N. H. Abd El-Wahab
    • 1
  • Ahmed Salah
    • 2
    Email author
  • A. S. Abdel Rady
    • 3
  • Abdel-Nasser A. Osman
    • 3
  1. 1.Mathematics Department, Faculty of ScienceMinia UniversityMiniaEgypt
  2. 2.Mathematics and Theoretical Physics Department, Nuclear Research CenterAtomic Energy AuthorityCairoEgypt
  3. 3.Mathematics Department, Faculty of ScienceSouth Valley UniversityQenaEgypt

Personalised recommendations